In order to fully understand how energy-eigenfunction shape depends on the potential energy function, we must introduce the Schrödinger equation, here written in terms of position variables,
[−(ħ^{2}/2m)∂^{2}/∂x^{2 }+ V(x)] ψ(x,t) = iħ(∂/∂t) ψ(x,t) , (1)
and the related time-independent Schrödinger equation (TISE),
[−(ħ^{2}/2m)d^{2}/dx^{2 }+ V(x)] ψ(x) = E ψ(x) , (2)
where ħ is Planck’s constant over 2π and V(x) is the potential energy function. Eq. (2) is also called an energy-eigenvalue equation since the solutions of the TISE are energy eigenstates with energy eigenfunctions, ψ(x), with energy E. For our purposes it is more convenient to write Eq. (2) as
(d^{2}/dx^{2}) ψ(x) = −(2m/ħ^{2}) [E − V(x)] ψ(x) . (3)
From Eq. (3), there are three general forms for energy eigenfunctions:
Where [E − V(x)] > 0, E > V(x) and the eigenfunction has an oscillatory form.
Where [E − V(x)] = 0, E = V(x) and the eigenfunction has a linear form.
Where [E − V(x)] < 0, E < V(x) and the eigenfunction has an exponential-type form which occurs in the classically forbidden region. This phenomenon is called tunneling.
The relationship between E and V(x) in a particular region is also important for determining energy-eigenfunction shape:
For [E − V(x)], the larger E > V(x) becomes, the curvier or wigglier the eigenfunction in this region. If V(x) changes, so does the curviness or wiggliness.^{17}
For [E − V(x)] < 0, the larger V(x) > E becomes, the stronger the exponential-type form (damped or rising) of the eigenfunction in this region. If, as in the case of the infinite square well, the potential energy is infinite, the amplitude of the energy eigenstate in this region is zero.
For a given potential energy function, in regions where E > V(x), a larger value of [E − V(x)], yields larger curviness (the state is more oscillatory) and from this one can determine the ordering of the energy eigenstates. The larger the value of [E − V(x)] (more curviness), the more zero crossings for a state, and a larger energy than that of a state with fewer zero crossings. In particular, since the ground state (n = 1) has no zero crossings, n = 1 + the number of zero crossings. Therefore, for a given well, a given energy eigenfunction will be curvier and have more zero crossings than one of less energy.
Note that we are consistently using the quantity [E − V(x)] and not calling it the kinetic energy. Calling the quantity [E − V(x)] the kinetic energy would be a mistake, because it would imply that we know the kinetic energy at a point in space, and therefore that we know the momentum at a point in space, p_{x}(x). This statement violates the Heisenberg uncertainty principle.